Taylor Series and Asymptotic Expansions
|
|
- Debra Clark
- 7 years ago
- Views:
Transcription
1 Taylor Series and Asymptotic Epansions The importance of power series as a convenient representation, as an approimation tool, as a tool for solving differential equations and so on, is pretty obvious. What may not be so obvious is that power series can be of some use even when they diverge! Let us start by considering Taylor series. If f : [ a,a] has infinitely many continuous derivatives, Taylor s theorem says that for each n IN {}, f( P n (+R n ( n with P n ( m m! f(m ( m R n ( (n+! f(n+ ( n+ for some between and Since f (n+ is continuous, there eists a constant M n such that R n ( M n n+ for all [ a,a]. This says that as gets smaller and smaller, the polynomial P n ( becomes a better and better approimation to f(. The rate at which P n ( tends to f( as tends to zero is, at worst, proportional to n+. Thus the Taylor series m m! f(m ( m is an asymptotic epansion for f( where Definition. m a m m is said to be an asymptotic epansion for f(, denoted f( m a m m if, for each n IN {}, lim n [f( n a m m] m There are several important observations to make about this definition. (i The definition says that, for each fied n, n m a m m becomes a better and better approimation to f( as gets smaller. As, n m a m m approaches f( faster than n tends to zero. (ii The definition says nothing about what happens as n. There is no guarantee that for each fied, n m a m m tends to f( as n. (iii In particular, the series m a m m may have radius of convergence zero. In this case, m a m m is just a formal symbol. It does not have any meaning as a series. (iv We have seen that every infinitely differentiable function has an asymptotic epansion, regardless of whether its Taylor series converges or not. Now back to our Taylor series. There are three possibilities. (i The series m m! f(m ( m has radius of convergence zero. (ii The series m m! f(m ( m has radius of convergence r > and lim R n( for all r < < r. n In this case the Taylor series converges to f( for all ( r,r. Then f( is said to be analytic on ( r,r. (iii The series m m! f(m ( m has radius of convergence r > and the Taylor series converges to something other than f(. Here is an eample of each of these three types of behaviour. Eample (i For this eample, f( e t + t dt. Our work on interchanging the order of differentiation and integration yields that f( is infinitely differentiable and that the derivatives are given by d n d n f( n n ( e t + t dt c Joel Feldman. 8. All rights reserved. February 8, 8 Taylor Series and Asymptotic Epansions
2 Instead of evaluating the derivative directly, we will use a little trick to generate the Taylor series for f(. Recall that r n r rn+ r We just substitute r t to give and f( n n N + t n ( t n + ( t N+ + t ( t n e t ( dt+ t N+ + t e t dt ( n n t n e t dt+( N+ (N+ n ( n n! n +( N+ (N+ n t N+ + t e t dt t N+ + t e t dt since ( t n e t dt n!. To compute the m th derivative, f (m (, we choose N such that (N + > m and apply { d m d n if m n m (n! if m n Applying dm d with m < (N + to ( N+ (N+ t N+ m + t e t dt gives a finite sum with every term containing a factor p with p >. When is set to, every such term is zero. So { d m d f( if m is odd m ( m!m! if m is even So the Taylor series epansion of f is m m! f(m ( m m m even ( m ( m! ( m m!m! m mn ( n n! n The high growth rate of n! makes the radius of convergence of this series zero. For eample, one formula for the radius of convergence of [ n a ]. [ ] n n is lim an+ n When an ( n n!, lim an+ n [ lim (n + ]. Replacing by gives that n n ( n n! n converges if and only if. Nonetheless, the error estimates N f( ( n n! n (N+ t N+ + t e t dt (N+ t N+ e t dt (N +! N+ n shows that if we fi N and make smaller and smaller, N n ( n n! n becomes a better and better approimation to f(. On the other hand, if we fi and let N grow, we see that the error grows dramatically for large N. Here is a graph of y (N+! N+ against N, for several different (fied values of. ( Denote γ n t n e t dt. Then γ e t dt and, by integration by parts with u t n, dv e t dt, du nt n dt and v e t, γ n nγ n for all n IN. So, by induction, γ n n!. c Joel Feldman. 8. All rights reserved. February 8, 8 Taylor Series and Asymptotic Epansions n
3 (N +! N N Eample (ii For this eample, f( e. By Taylor s theorem, f( n m m m! +R n( with R n ( e (n+! n+ for some between and. In this case, m m m! has radius of convergence infinity and the error Rn ( n! e n tends to zero both (a when n is fied and tends to zero and (b when is fied and n tends to infinity. Use R n ( n! e n to denote our bound on the error introduced by truncating the series at n n!. Then R n+ ( R n+ n (. So if, for eample, n, then increasing the inde n of R n ( by one reduces the value of R n ( by a factor of at least two. Eample (iii For this eample, f( { e / if if I shall show below that f is infinitely differentiable and f (n ( for every n IN {}. So in this case, the Taylor series is n n and has infinite radius of convergence. When we approimate f( by the Taylor polynomial, n m m, of degree n, the error, R n ( f(, is independent of n. The function f( agrees with its Taylor series only for. Similarly, the function g( { e / if > if agrees with its Taylor series if and only if. Here is the argument that f is infinitely differentiable and f (n ( for every n IN {}. For any and any integer n ( e / + + (! + + n ( n! + n n! Consequently, for any polynomial P(y, say of degree p, there is a constant C such that ( P e / P ( e C p / (p+!c +p (p+! (p+ c Joel Feldman. 8. All rights reserved. February 8, 8 Taylor Series and Asymptotic Epansions
4 for all. Hence, for any polynomial P(y, Applying ( with P(y y, we have lim P( e /. ( f f( f( ( lim lim e / f ( Applying ( with P(y y 4, we have f f ( lim ( f ( lim ( e / f ( { if e / if { if ( e / if 6 ( Indeed, for any n, if f (n ( P n e / for all >, then applying ( with P(y yp n (y gives f (n f ( lim (n ( f (n ( lim P n ( e / and hence { if f (n ( [ P ( n ( + P n ] e / if Eample (iv Stirling formula lnn! ( n+ lnn n+ln π actually consists of the first few terms of an asymptotic epansion lnn! ( n+ lnn n+ln π+ n 6 n + If n >, the approimation lnn! ( n+ lnn n+ln π is accurate to within.6% and the eponentiated form n! n n+ πe n+ n is accurate to one part in,. But fiing n and taking many more terms in the epansion will in fact give you a much worse approimation to lnn!. Laplace s Method Stirling s formula may be generated by Laplace s method one of the more common ways of generating asymptotic epansions. Laplace s method provides good approimations to integrals of the form I(s e sf( d when s is very large and the function f takes its minimum value at a unique point, m. The method is based on the following three observations. (i When s is verylargeand m, e sf( e sf(m e s[f( f(m] e sf(m, since f( f( m >. So, for large s, the integral will be dominated by values of near m. (ii For near m f( f( m + f ( m ( m since f ( m. (iii Integrals of the form e a( m d, or more generally e a( m ( m n d, can be computed eactly. Let s implement this strategy for the Γ function, defined in the lemma below. This will give Stirling s formula, because of part (d of the lemma. c Joel Feldman. 8. All rights reserved. February 8, 8 Taylor Series and Asymptotic Epansions 4
5 Lemma. Define, for s >, the Gamma function Γ(s t s e t dt (a Γ( (b Γ ( π. (c For any s >, Γ(s+ sγ(s. (d For any n IN, Γ(n+ n!. Proof: (a is a trivial integration. (b Make the change of variables t, dt d td. This gives Γ ( t e t dt We compute the square of this integral by switching e d e d Γ ( ( ( e d e y dy e ( +y ddy IR to polar coordinates. This standard trick gives Γ ( π dθ dr re r π dθ [ e r] π dθ π (c By by integration by parts with u t s, dv e t dt, du st s dt and v e t Γ(s+ t s e t dt t s ( e t ( e t st s dt s t s e t dt sγ(s (d is obvious by induction, using parts (a and (c. To derive Stirling s formula, using Laplace s method, We first manipulate Γ(s + into the form e sf( d. Γ(s+ t s e t dt s s+ s e s d s s+ e s e s( ln d where t s So f( ln and f (. Consequently f ( < for < and f ( > for >, so that f has a unique minimum at and increases monotonically as gets farther from. The minimum value of f is f(. (We multiplied and divided by e s in the last line above in order to arrange that the minimum value of f be eactly zero. c Joel Feldman. 8. All rights reserved. February 8, 8 Taylor Series and Asymptotic Epansions 5
6 If ε > is any fied positive number, e sf( d +ε ε e sf( d will be agoodapproimationfor anysufficiently larges, because, when, f( > and lim s e sf(. We will derive, in an appendi, eplicit bounds on the error introduced by this approimation as well as by the other approimations we make in the course of this development. The remaining couple of steps in the procedure are motivated by the fact that integrals of the form e a( m ( m n d can be computed eactly. If we have chosen ε pretty small, Taylor s formula f( f(+f (( + f( (( +! f( (( + 4! f(4 (( 4 [ [ [ ] ( +! ] ( + ] 4! 4 ( 4 ( ( + 4 ( 4 will give a good approimation for all < ε, so that e sf( d +ε ε e s[ ( ( + 4 ( 4] d +ε ε e s ( e [s ( s 4 ( 4] d (We shall find the first two terms an epansion for Γ(s+. If we wanted to find more terms, we would keep more terms in the Taylor epansion of f(. The net step is to Taylor epand e [s ( s 4 ( 4] about, in order to give integrands of the form e s ( ( n. We could apply Taylor s formula directly, but it is easier to use the known epansion of the eponential. e sf( d +ε ε +ε ε e s ( { + [ s ( s 4 ( 4] + [ s ( s 4 ( 4] } d e s ( { + s ( s 4 ( 4 + s 8 ( 6} d e s ( { + s ( s 4 ( 4 + s 8 ( 6} d e s ( { s 4 ( 4 + s 8 ( 6} d since ( e s ( is odd about and hence integrates to zero. For n an even natural number e s ( ( n d ( s n+ ( s n+ ( s n+ ( s n+ Γ ( n+ e y y n dy where y e y y n dy e t t n dt s ( where t y, dt ydy tdy ( By the lemma at the beginning of this section, Γ ( ( π Γ Γ( ( π Γ 5 Γ( π Γ ( 7 5 Γ( 5 5 c Joel Feldman. 8. All rights reserved. February 8, 8 Taylor Series and Asymptotic Epansions 6 π
7 so that e s( ln d ( s Γ ( ( π s s 5 4( Γ ( 5 s s 8 85 s s π ( + 4s + 5 6s + s 8( s 7 Γ ( and Γ(s+ s ( + s+ e s π + s So we have the first two terms in the asymptotic epansion for Γ. We can obviously get as many as we like by taking more and more terms in the Taylor epansion for f( and e y. So far, we have not attempted to keep track of the errors introduced by the various approimations. It is not hard to do so, but it is messy. So we have relegated the error estimates to the appendi. Summability We have now seen one way in which divergent epansions may be of some use. There is another possibility. Even if we cannot cannot take the sum of the series n in the conventional sense, there may be some more general sense in which the series is summable. We will just briefly take a look at two such generalizations. Suppose that n is a (possibly divergent series. Let S N denote the series partial sum. Then the series is said to converge in the sense of Cesàro if the limit n S S c lim +S + +S N N N+ eists. That is, if the limit of the average of the partial sums eists. The series is said to converge in the sense of Borel if the power series g(t has radius of convergence and the integral S B n n! t n e t g(t dt converges. As the following theorem shows, these two definitions each etends our usual definition of series convergence. Theorem. If the series n converges, then S C and S B both eist and S C S B n Proof that S C n. Write S n and σ N S+S+ +SN N+ and observe that S σ N (S S +(S S + +(S S N N + c Joel Feldman. 8. All rights reserved. February 8, 8 Taylor Series and Asymptotic Epansions 7
8 Pick any ε >. Since S lim S n there is an M such that S S m < ε n for all m M. Also since S lim S n, the sequence { } S n is bounded. That is, there is a K such that Sn K for all n. So if n N > ma { MK ε/,m} S σn S S + S S + + S S N N + S S + S S + + S S M + S S M + S S M+ + + S S N N + N + ε ε ε K + K + + K < + N + N + ε + ε ε MK N + Hence lim N σ N S and S C S. Proof that S B n. Write S n n The last step, namely the echange n n! n! + n! n n e t t n dt N M + N + g(t ε { }} { n! t n e t dt n can be justified rigorously. If the radius of convergence of the series n α n is strictly greater than one, the justification is not very hard. But in general it is too involved to give here. That s why Proof is in quotation marks. The converse of this theorem is false of course. Consider, for eample, n ( rn with r. Then, as n, n S n ( r m ( rn+ converges to +r σ N N+ m if r < and diverges if r. But S n n +r + N+ +r r N+ (r+ + N+ As N, this converges to +r g(t n n +r [ N ( r n+] ( r N+ (r+ n N+ +r if r and diverges if r >. And ( r n n! t n e rt e t g(t dt ] [N + ( r ( rn+ +r e (+rt dt +r if r > and diverges otherwise. Collecting together these three results, in the conventional sense if r (, ( r n +r in the sense of Cesàro if r (,] n in the sense of Borel if r (, c Joel Feldman. 8. All rights reserved. February 8, 8 Taylor Series and Asymptotic Epansions 8
9 By way of conclusion, let me remark that these generalized modes of summation are not just mathematician s toys. The Fourier series of any continuous function converges in the sense of Cesàro. To get conventional convergence, one needs a certain amount of smoothness in addition of continuity. Also the calculation of the Lamb shift of the hydrogen atom spectrum, which provided (in addition to a Nobel prize spectacular numerical agreement with one of the most accurate measurements in science, consisted of adding up the first few terms of a series. The rigorous status of this series is not known. But umber of similar series (arising in other quantum field theories are known to diverge in the conventional sense but to converge in the sense of Borel. Appendi - Error Terms for Stirling s Formula In this appendi, we will treat the error terms arising in our development of Stirling s formula. A precise statement of that development is Step. Recall that f( ln. Write e sf( d and discard the last two terms. Step. Write +ε ε +ε ε e sf( d e sf( d+ +ε ε ε e sf( d+ e sf( d +ε e s( e s[f( ( ] d Apply e t +t+ t +e c! t with t s [ f( ( ] to give e s[f( ( ] s [ f( ( ] + s[ f( ( ] e c! s[ f( ( ] where c is between and s [ f( ( ]. Discard the last term. Step. Taylor epand f( ( ( + 4 ( 4 5 ( 5 + f(6 (c 6! ( 6 for some c [ ε,+ε], substitute this into s [ f( ( ] + s[ f( ( ] and discard any terms that contain f (6 (c and any terms of the form s m ( n with m n+ 5. (You ll see the reason for this condition shortly. This leaves +ε ε because the terms with n odd integrate to zero. Step 4. Replace +ε ε by The errors introduced in each step are as follows: Step. ε e sf( d and +ε e sf( d e s( { s 4 ( 4 + s 8 ( 6} d Step. +ε ε e s( e c! s[ f( ( ] d Step. A finite number of terms, each of the form umerical constant times an integral +ε ε e s( s m ( n d with m n+ 5. Here we have used that, since c [ ε,+ε], there is a constant M such that f (6 (c M. c Joel Feldman. 8. All rights reserved. February 8, 8 Taylor Series and Asymptotic Epansions 9
10 Step 4. A finite number of terms, each of the form umerical constant times an integral ε e s( s m ( n d. To treat these error terms, we use the following information about f( ln. (F f increases monotonically as increases. (F We have fied some small ε >. There is a δ > such that f( δ( for all +ε. To see that this is the case, we first observe that if δ ( d d f( δ( δ. So it suffices to pick δ sufficiently small that +ε δ The latter is always possible just because f(+ε >. (F For each [ ε,+ε] there is a c [ ε,+ε] such that Hence, there is a constant M such that f( (! f( (c ( f( ( M for all [ ε,+ε]. (F4 If [ ε,+ε] with ε small enough that εm 4, f( ( M 4 ( We can now get upper bounds on each of the error terms. For step, we have ε e sf( d F e sf( ε ε d e sf( ε and that f(+ε δε. and For step, we have +ε e sf( d F +ε e sδ( d δs e δεs +ε ε For step, we have e s( e c! s f( ( d F4 +ε ε e s ( s m n d! s +ε ε e s ( e s 4 ( f( ( d F +ε! M s e s 4 ( 9 d ε! M s e s 4 ( 9 d! M s ( s 4 5Γ(5 M s as in ( e s ( s m n d n+ s m( s M s 5 Γ ( n+ as in ( c Joel Feldman. 8. All rights reserved. February 8, 8 Taylor Series and Asymptotic Epansions
11 if m n+ 5. For step 4, we have e s ( s m n d ε ε e 4 ε s e 4 ε s e s 4 ( e s 4 ( s m n d ε e 4 εs s m( s 4 n+ M s m n+ e 4 ε s e s 4 ( s m n d e s 4 ( s m n d Γ ( n+ as in ( As promised these error terms all decay faster than πs ( + s as s. In particular, error terms introduced in Steps and 4, by ignoring contributions for ε, decay eponentially as s. Error terms introduced in Steps and, by dropping part of the Taylor series approimation, decay like s p for some p, depending on how much of the Taylor series we discarded. In particular, error terms introduced in Step decay like s 5, because of the condition m n+ 5. c Joel Feldman. 8. All rights reserved. February 8, 8 Taylor Series and Asymptotic Epansions
About the Gamma Function
About the Gamma Function Notes for Honors Calculus II, Originally Prepared in Spring 995 Basic Facts about the Gamma Function The Gamma function is defined by the improper integral Γ) = The integral is
More information10.2 Series and Convergence
10.2 Series and Convergence Write sums using sigma notation Find the partial sums of series and determine convergence or divergence of infinite series Find the N th partial sums of geometric series and
More informationn k=1 k=0 1/k! = e. Example 6.4. The series 1/k 2 converges in R. Indeed, if s n = n then k=1 1/k, then s 2n s n = 1 n + 1 +...
6 Series We call a normed space (X, ) a Banach space provided that every Cauchy sequence (x n ) in X converges. For example, R with the norm = is an example of Banach space. Now let (x n ) be a sequence
More informationStirling s formula, n-spheres and the Gamma Function
Stirling s formula, n-spheres and the Gamma Function We start by noticing that and hence x n e x dx lim a 1 ( 1 n n a n n! e ax dx lim a 1 ( 1 n n a n a 1 x n e x dx (1 Let us make a remark in passing.
More informationSection 1.3 P 1 = 1 2. = 1 4 2 8. P n = 1 P 3 = Continuing in this fashion, it should seem reasonable that, for any n = 1, 2, 3,..., = 1 2 4.
Difference Equations to Differential Equations Section. The Sum of a Sequence This section considers the problem of adding together the terms of a sequence. Of course, this is a problem only if more than
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationRepresentation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
More information1 Error in Euler s Method
1 Error in Euler s Method Experience with Euler s 1 method raises some interesting questions about numerical approximations for the solutions of differential equations. 1. What determines the amount of
More informationsin(x) < x sin(x) x < tan(x) sin(x) x cos(x) 1 < sin(x) sin(x) 1 < 1 cos(x) 1 cos(x) = 1 cos2 (x) 1 + cos(x) = sin2 (x) 1 < x 2
. Problem Show that using an ɛ δ proof. sin() lim = 0 Solution: One can see that the following inequalities are true for values close to zero, both positive and negative. This in turn implies that On the
More informationRandom variables, probability distributions, binomial random variable
Week 4 lecture notes. WEEK 4 page 1 Random variables, probability distributions, binomial random variable Eample 1 : Consider the eperiment of flipping a fair coin three times. The number of tails that
More informationSequences and Series
Sequences and Series Consider the following sum: 2 + 4 + 8 + 6 + + 2 i + The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite
More informationHomework 2 Solutions
Homework Solutions 1. (a) Find the area of a regular heagon inscribed in a circle of radius 1. Then, find the area of a regular heagon circumscribed about a circle of radius 1. Use these calculations to
More information6. Define log(z) so that π < I log(z) π. Discuss the identities e log(z) = z and log(e w ) = w.
hapter omplex integration. omplex number quiz. Simplify 3+4i. 2. Simplify 3+4i. 3. Find the cube roots of. 4. Here are some identities for complex conjugate. Which ones need correction? z + w = z + w,
More informationCOMP 250 Fall 2012 lecture 2 binary representations Sept. 11, 2012
Binary numbers The reason humans represent numbers using decimal (the ten digits from 0,1,... 9) is that we have ten fingers. There is no other reason than that. There is nothing special otherwise about
More informationParabolic Equations. Chapter 5. Contents. 5.1.2 Well-Posed Initial-Boundary Value Problem. 5.1.3 Time Irreversibility of the Heat Equation
7 5.1 Definitions Properties Chapter 5 Parabolic Equations Note that we require the solution u(, t bounded in R n for all t. In particular we assume that the boundedness of the smooth function u at infinity
More informationLectures 5-6: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
More informationMoving Least Squares Approximation
Chapter 7 Moving Least Squares Approimation An alternative to radial basis function interpolation and approimation is the so-called moving least squares method. As we will see below, in this method the
More informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
More informationMATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
More informationTHE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION
THE SINE PRODUCT FORMULA AND THE GAMMA FUNCTION ERICA CHAN DECEMBER 2, 2006 Abstract. The function sin is very important in mathematics and has many applications. In addition to its series epansion, it
More informationChapter 4. Polynomial and Rational Functions. 4.1 Polynomial Functions and Their Graphs
Chapter 4. Polynomial and Rational Functions 4.1 Polynomial Functions and Their Graphs A polynomial function of degree n is a function of the form P = a n n + a n 1 n 1 + + a 2 2 + a 1 + a 0 Where a s
More informationTriangle deletion. Ernie Croot. February 3, 2010
Triangle deletion Ernie Croot February 3, 2010 1 Introduction The purpose of this note is to give an intuitive outline of the triangle deletion theorem of Ruzsa and Szemerédi, which says that if G = (V,
More informationCONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
More informationMathematical Induction
Mathematical Induction In logic, we often want to prove that every member of an infinite set has some feature. E.g., we would like to show: N 1 : is a number 1 : has the feature Φ ( x)(n 1 x! 1 x) How
More informationAn Introduction to Partial Differential Equations in the Undergraduate Curriculum
An Introduction to Partial Differential Equations in the Undergraduate Curriculum J. Tolosa & M. Vajiac LECTURE 11 Laplace s Equation in a Disk 11.1. Outline of Lecture The Laplacian in Polar Coordinates
More informationStanford Math Circle: Sunday, May 9, 2010 Square-Triangular Numbers, Pell s Equation, and Continued Fractions
Stanford Math Circle: Sunday, May 9, 00 Square-Triangular Numbers, Pell s Equation, and Continued Fractions Recall that triangular numbers are numbers of the form T m = numbers that can be arranged in
More informationContinued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
More informationMATH 4330/5330, Fourier Analysis Section 11, The Discrete Fourier Transform
MATH 433/533, Fourier Analysis Section 11, The Discrete Fourier Transform Now, instead of considering functions defined on a continuous domain, like the interval [, 1) or the whole real line R, we wish
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 5 9/17/2008 RANDOM VARIABLES Contents 1. Random variables and measurable functions 2. Cumulative distribution functions 3. Discrete
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More informationMATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
More informationStudent Performance Q&A:
Student Performance Q&A: 2008 AP Calculus AB and Calculus BC Free-Response Questions The following comments on the 2008 free-response questions for AP Calculus AB and Calculus BC were written by the Chief
More informationMathematics 31 Pre-calculus and Limits
Mathematics 31 Pre-calculus and Limits Overview After completing this section, students will be epected to have acquired reliability and fluency in the algebraic skills of factoring, operations with radicals
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationSample Induction Proofs
Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given
More informationLies My Calculator and Computer Told Me
Lies My Calculator and Computer Told Me 2 LIES MY CALCULATOR AND COMPUTER TOLD ME Lies My Calculator and Computer Told Me See Section.4 for a discussion of graphing calculators and computers with graphing
More informationI. GROUPS: BASIC DEFINITIONS AND EXAMPLES
I GROUPS: BASIC DEFINITIONS AND EXAMPLES Definition 1: An operation on a set G is a function : G G G Definition 2: A group is a set G which is equipped with an operation and a special element e G, called
More informationAP Calculus BC 2008 Scoring Guidelines
AP Calculus BC 8 Scoring Guidelines The College Board: Connecting Students to College Success The College Board is a not-for-profit membership association whose mission is to connect students to college
More informationf x a 0 n 1 a 0 a 1 cos x a 2 cos 2x a 3 cos 3x b 1 sin x b 2 sin 2x b 3 sin 3x a n cos nx b n sin nx n 1 f x dx y
Fourier Series When the French mathematician Joseph Fourier (768 83) was tring to solve a problem in heat conduction, he needed to epress a function f as an infinite series of sine and cosine functions:
More informationMath 2443, Section 16.3
Math 44, Section 6. Review These notes will supplement not replace) the lectures based on Section 6. Section 6. i) ouble integrals over general regions: We defined double integrals over rectangles in the
More informationDiscrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2
CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 2 Proofs Intuitively, the concept of proof should already be familiar We all like to assert things, and few of us
More informationA simple criterion on degree sequences of graphs
Discrete Applied Mathematics 156 (2008) 3513 3517 Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: www.elsevier.com/locate/dam Note A simple criterion on degree
More informationUndergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
More informationLemma 5.2. Let S be a set. (1) Let f and g be two permutations of S. Then the composition of f and g is a permutation of S.
Definition 51 Let S be a set bijection f : S S 5 Permutation groups A permutation of S is simply a Lemma 52 Let S be a set (1) Let f and g be two permutations of S Then the composition of f and g is a
More informationFollow links for Class Use and other Permissions. For more information send email to: permissions@pupress.princeton.edu
COPYRIGHT NOTICE: Ariel Rubinstein: Lecture Notes in Microeconomic Theory is published by Princeton University Press and copyrighted, c 2006, by Princeton University Press. All rights reserved. No part
More informationMathematical Induction. Lecture 10-11
Mathematical Induction Lecture 10-11 Menu Mathematical Induction Strong Induction Recursive Definitions Structural Induction Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More informationIntegrating algebraic fractions
Integrating algebraic fractions Sometimes the integral of an algebraic fraction can be found by first epressing the algebraic fraction as the sum of its partial fractions. In this unit we will illustrate
More informationCore Maths C1. Revision Notes
Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the
More informationLS.6 Solution Matrices
LS.6 Solution Matrices In the literature, solutions to linear systems often are expressed using square matrices rather than vectors. You need to get used to the terminology. As before, we state the definitions
More informationEvery Positive Integer is the Sum of Four Squares! (and other exciting problems)
Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer
More information6.2 Permutations continued
6.2 Permutations continued Theorem A permutation on a finite set A is either a cycle or can be expressed as a product (composition of disjoint cycles. Proof is by (strong induction on the number, r, of
More informationProbability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce
More informationNotes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.
More informationNote on some explicit formulae for twin prime counting function
Notes on Number Theory and Discrete Mathematics Vol. 9, 03, No., 43 48 Note on some explicit formulae for twin prime counting function Mladen Vassilev-Missana 5 V. Hugo Str., 4 Sofia, Bulgaria e-mail:
More informationTaylor Polynomials. for each dollar that you invest, giving you an 11% profit.
Taylor Polynomials Question A broker offers you bonds at 90% of their face value. When you cash them in later at their full face value, what percentage profit will you make? Answer The answer is not 0%.
More informationA power series about x = a is the series of the form
POWER SERIES AND THE USES OF POWER SERIES Elizabeth Wood Now we are finally going to start working with a topic that uses all of the information from the previous topics. The topic that we are going to
More informationPYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationChapter 4 One Dimensional Kinematics
Chapter 4 One Dimensional Kinematics 41 Introduction 1 4 Position, Time Interval, Displacement 41 Position 4 Time Interval 43 Displacement 43 Velocity 3 431 Average Velocity 3 433 Instantaneous Velocity
More informationAN ANALYSIS OF A WAR-LIKE CARD GAME. Introduction
AN ANALYSIS OF A WAR-LIKE CARD GAME BORIS ALEXEEV AND JACOB TSIMERMAN Abstract. In his book Mathematical Mind-Benders, Peter Winkler poses the following open problem, originally due to the first author:
More informationPartial Fractions Decomposition
Partial Fractions Decomposition Dr. Philippe B. Laval Kennesaw State University August 6, 008 Abstract This handout describes partial fractions decomposition and how it can be used when integrating rational
More information3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
More informationThe small increase in x is. and the corresponding increase in y is. Therefore
Differentials For a while now, we have been using the notation dy to mean the derivative of y with respect to. Here is any variable, and y is a variable whose value depends on. One of the reasons that
More information5544 = 2 2772 = 2 2 1386 = 2 2 2 693. Now we have to find a divisor of 693. We can try 3, and 693 = 3 231,and we keep dividing by 3 to get: 1
MATH 13150: Freshman Seminar Unit 8 1. Prime numbers 1.1. Primes. A number bigger than 1 is called prime if its only divisors are 1 and itself. For example, 3 is prime because the only numbers dividing
More informationSolutions to Homework 10
Solutions to Homework 1 Section 7., exercise # 1 (b,d): (b) Compute the value of R f dv, where f(x, y) = y/x and R = [1, 3] [, 4]. Solution: Since f is continuous over R, f is integrable over R. Let x
More informationGeneral Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1
A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions
More informationRoots of Equations (Chapters 5 and 6)
Roots of Equations (Chapters 5 and 6) Problem: given f() = 0, find. In general, f() can be any function. For some forms of f(), analytical solutions are available. However, for other functions, we have
More informationTransformations and Expectations of random variables
Transformations and Epectations of random variables X F X (): a random variable X distributed with CDF F X. Any function Y = g(x) is also a random variable. If both X, and Y are continuous random variables,
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationMathematical Methods of Engineering Analysis
Mathematical Methods of Engineering Analysis Erhan Çinlar Robert J. Vanderbei February 2, 2000 Contents Sets and Functions 1 1 Sets................................... 1 Subsets.............................
More informationMathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
More informationExponential Functions, Logarithms, and e
chapter 3 Starry Night, painted by Vincent Van Gogh in 889. The brightness of a star as seen from Earth is measured using a logarithmic scale. Eponential Functions, Logarithms, and e This chapter focuses
More informationMATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
More informationNotes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
More informationWHERE DOES THE 10% CONDITION COME FROM?
1 WHERE DOES THE 10% CONDITION COME FROM? The text has mentioned The 10% Condition (at least) twice so far: p. 407 Bernoulli trials must be independent. If that assumption is violated, it is still okay
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More informationChapter 31 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M.
31 Geometric Series Motivation (I hope) Geometric series are a basic artifact of algebra that everyone should know. 1 I am teaching them here because they come up remarkably often with Markov chains. The
More informationTHE CENTRAL LIMIT THEOREM TORONTO
THE CENTRAL LIMIT THEOREM DANIEL RÜDT UNIVERSITY OF TORONTO MARCH, 2010 Contents 1 Introduction 1 2 Mathematical Background 3 3 The Central Limit Theorem 4 4 Examples 4 4.1 Roulette......................................
More informationECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE
ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.
More informationPartial Fractions Examples
Partial Fractions Examples Partial fractions is the name given to a technique of integration that may be used to integrate any ratio of polynomials. A ratio of polynomials is called a rational function.
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by Men-Gen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationScalar Valued Functions of Several Variables; the Gradient Vector
Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x = φ(x 1,
More informationNumerical Methods for Option Pricing
Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly
More informationCS 103X: Discrete Structures Homework Assignment 3 Solutions
CS 103X: Discrete Structures Homework Assignment 3 s Exercise 1 (20 points). On well-ordering and induction: (a) Prove the induction principle from the well-ordering principle. (b) Prove the well-ordering
More informationThe Heat Equation. Lectures INF2320 p. 1/88
The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)
More informationReview Solutions MAT V1102. 1. (a) If u = 4 x, then du = dx. Hence, substitution implies 1. dx = du = 2 u + C = 2 4 x + C.
Review Solutions MAT V. (a) If u 4 x, then du dx. Hence, substitution implies dx du u + C 4 x + C. 4 x u (b) If u e t + e t, then du (e t e t )dt. Thus, by substitution, we have e t e t dt e t + e t u
More information9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
More informationThe numerical values that you find are called the solutions of the equation.
Appendi F: Solving Equations The goal of solving equations When you are trying to solve an equation like: = 4, you are trying to determine all of the numerical values of that you could plug into that equation.
More informationG. GRAPHING FUNCTIONS
G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression
More informationMath 21a Curl and Divergence Spring, 2009. 1 Define the operator (pronounced del ) by. = i
Math 21a url and ivergence Spring, 29 1 efine the operator (pronounced del by = i j y k z Notice that the gradient f (or also grad f is just applied to f (a We define the divergence of a vector field F,
More information2008 AP Calculus AB Multiple Choice Exam
008 AP Multiple Choice Eam Name 008 AP Calculus AB Multiple Choice Eam Section No Calculator Active AP Calculus 008 Multiple Choice 008 AP Calculus AB Multiple Choice Eam Section Calculator Active AP Calculus
More informationSECTION 10-2 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
More informationSOLUTIONS TO EXERCISES FOR. MATHEMATICS 205A Part 3. Spaces with special properties
SOLUTIONS TO EXERCISES FOR MATHEMATICS 205A Part 3 Fall 2008 III. Spaces with special properties III.1 : Compact spaces I Problems from Munkres, 26, pp. 170 172 3. Show that a finite union of compact subspaces
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More information3 Contour integrals and Cauchy s Theorem
3 ontour integrals and auchy s Theorem 3. Line integrals of complex functions Our goal here will be to discuss integration of complex functions = u + iv, with particular regard to analytic functions. Of
More informationWalrasian Demand. u(x) where B(p, w) = {x R n + : p x w}.
Walrasian Demand Econ 2100 Fall 2015 Lecture 5, September 16 Outline 1 Walrasian Demand 2 Properties of Walrasian Demand 3 An Optimization Recipe 4 First and Second Order Conditions Definition Walrasian
More information88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a
88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small
More informationLecture 13 - Basic Number Theory.
Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted
More information